Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-31
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FALSE: the Radon-Nikodym derivative is a uniquely determined function

Statement

Assume the Axiom of Countable Choice.

False claim: For ν≪μ, the derivative dν/dμ is a uniquely determined pointwise function.

Facts & Assumptions

Given: Countable choice, the zero measure, and the Cantor set C.

[L1]

The Cantor set is Lebesgue null. (The Cantor set is an uncountable subset of R of Lebesgue measure zero)

[L2]

The integral over a null set vanishes. (A nonnegative integral over a null set vanishes)

[L3]

The Radon-Nikodym derivative is only an almost-everywhere equivalence class. (The Radon-Nikodym derivative as an almost-everywhere equivalence class)

Refutation

technique · direct
1.1

The functions 0 and χC both integrate to 0 over every measurable set by [L1] and [L2], so they both represent the Radon-Nikodym derivative of the zero measure with respect to Lebesgue measure.

L1L2given
2.1

These two representing functions are not equal pointwise, while [L3] allows exactly this kind of null-set discrepancy. Hence pointwise uniqueness is false.

step 1.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources